Slant lens interlacing with linearly arranged sets of lenses

ABSTRACT

An optical product that includes a transparent lens sheet, which has a first side with a plurality of side-by-side sets of linearly arranged lenses. Each of the sets of lenses is at a slant angle in the range of 10 to 46 degrees from a vertical or a horizontal axis of the lens sheet. The product includes an image layer that includes pixels from a number of digital images. The pixels are arranged in a pattern of pixel locations providing non-orthogonal interlacing of the digital images relative to each of the sets of the linearly arranged lenses. The pattern of pixel locations aligns a number of the pixels from each of the digital images to be parallel to a line extending through a center of the linearly arranged lenses in each set. Each of the linearly arranged lenses may have a round base, a hexagonal base, or a square base.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of U.S. patent applicationSer. No. 14/088,519, filed Nov. 25, 2013, which claims the benefit ofU.S. Provisional Application No. 61/797,145, filed Nov. 30, 2012, bothof which are incorporated herein by reference in their entireties.

BACKGROUND

1. Field of the Description

This description is generally directed toward methods of interlacingimages for use in printing images viewable through a lenticular lensarray or lens sheet, and, more particularly, to methods of interlacingto provide an increased amount of information (e.g., interlaced imagesor frames) underneath each lenticule to facilitate use of thinner lenssheets.

2. Relevant Background

Elaborate graphics or visual displays can be produced through the use ofsheets of lenticular lens arrays as these arrays of lenses can becombined with printed interlaced images to provide three-dimensional(3D) and animated imagery. For example, lenticular lens material is usedin the packaging industry for creating promotional material withappealing graphics and typically involves producing a sheet oflenticular lens material and adhesively attaching the lenticular lensmaterial to a separately produced object for display. The production oflenticular lenses is well known and described in detail in a number ofU.S. patents, including U.S. Pat. No. 5,967,032 to Bravenec et al. andU.S. Pat. No. 6,781,761 to Raymond.

In general, the production process includes selecting segments fromvisual images to create a desired visual effect, slicing each segmentinto a predefined number of slices or elements (such as 10 to 30 or moreslices per segment), and interlacing the segments and their slices(i.e., planning the layout of the numerous images). Lenticular lenses orlens sheets are then fabricated according to the number of slices or theinterlacing may be performed to suit the lens sheets, e.g., to suit aparticular lenticules or lenses per inch (LPI) of the lens sheet. Thelenticular lenses generally include a transparent web that has a flatside or layer and a side with optical ridges and grooves formed bylinear or elongated lenticules (i.e., lenses) arranged side-by-side withthe lenticules or optical ridges extending parallel to each other overthe length of the transparent web. To provide the unique visual effects,ink (e.g., four color ink) is applied to or printed directly on the flatside of the transparent web to form a thin ink layer (or a printed imageis applied with adhesive to the back or planar side of the transparentweb), which is then viewable through the transparent web of opticalridges.

Each lenticule or lens of the lenticular layer is paired or mapped to aset or number of the interlaced image slices or elements. Generally,only one of the slices is visible through the lenticule at a time basedon the position of the lenticule relative to a viewer's eye. In otherwords, the animation, 3D, or other graphic effect is achieved by movingthe lenticule or the viewer's position to sequentially view each of theinterlaced image slices under the lenticule and allow a viewer to seeeach segment of the image by combining the slices viewed from all thelenticules.

In producing conventional lenticular lens material, it is desirable touse as little material as possible, i.e., to produce effectivelenticules or lenticular lens arrays with as thin web material aspossible. Decreasing lens thickness is also desirable to facilitatefabrication using techniques such as web printing that are verydifficult or impractical with thicker lens materials. Thin lenticularlens material is desired to save on material costs and to provide arelatively flexible lens material or substrate that can be easilyapplied to products and product containers, such as in a label that canbe attached to a box or to a bottle as part of a wraparound label or ona cup to provide desirable visual effects. To make lenticular lensmaterials thinner, the whole structure must be properly scaled downwardtogether. In other words, the lenticules and the printed interlacedimage must be shrunk or made smaller together to allow proper mapping ofthe image slices to the lenticules.

However, such shrinking of the lenticules has proven very difficult withlimitations associated with printing the interlaced images oftenpreventing the lens layer or web from being made very thin. As notedabove, all the interlaced slices for each segment are placed underneatha single lenticule such that numerous slices have to be printed withvery little width to be mapped to the lenticules width or pitch. Withcoarser lens arrays (i.e., with lower the frequency or LPI), theprinting can be accomplished more easily and mapping to lenticules ofthe image slices achieved more accurately. However, coarser lens arrayswith frequencies of 10 to 30 LPI tend to be very thick because generalphysics or optical rules for focusing with conventional lenticularmaterial require that more lens thickness or more lens material beprovided to achieve effective focusing. For example, a 15 LPI lenticularlens array with a fairly common viewing angle (such as a 22-degreeviewing angle) may be mapped to an interlaced image that is printed orprovided directly behind the lenticular lens array, with each of thelenticules in the lens array being mapped to or paired with all imageslices of a paired segment of the interlaced image. If the lens array isformed from acrylic, the lens array would need to be about ⅜-inch thickto enable the lenticules to properly focus on the paired image slices.

Traditionally, lenticular printing has been a multi-step process thatincludes creating a lenticular image from at least two images andcombining it with a lenticular lens. The lenticular printing process canbe used to create various frames of animation for a motion effect, canbe used for offsetting the various layers at different increments for a3D effect, or can be used simply to show a set of alternate images thatmay appear to transform into each other. Once the various images arecollected, they are flattened into individual, different frame files,and, then, the frame files are digitally combined into a single finalfile for use in printing an interlaced image. The digital combiningprocess is often called “interlacing.”

Once the combined or interlaced file is generated, it can be used toprint an interlaced image directly to the back (or smooth/planar) sideof the lenticular lens sheet. In other applications, the interlacedimage may be printed onto a substrate (e.g., a synthetic paper or thelike), which is then laminated onto the lens (e.g., a transparentadhesive may be used to attach the substrate with the printed interlacedimage onto the lenticular lens sheet). When printing to the backside ofthe lens sheet, the registration of the thin slices or elongatedinterlaced images to the lenses is important during the lithographic orscreen printing process to avoid or at least limit ghosting or othereffects that produce poor imagery.

With traditional lenticular interlacing, each image is arranged orsliced into strips, which are then interlaced with one or more similarlyarranged or sliced images such as by splicing or interlacing. The endresult is that a person's single eye looking at the printed interlacedimage through the lenticular lens array (or lens sheet) sees a singlewhole image while a person's two eyes may see different images (e.g.,right and left-eye images), which provides a desired autostereoscopic or3D perception.

The process of creating strips of information from graphics or imagesand then scrambling them into a single image for printing underneath alens sheet can be problematic. One significant problem is that there isa limitation on the amount of information (e.g., pixels) that can beplaced underneath each lenticule or elongated lens in the lens sheet.For example, a lens or lenticule has a particular size (e.g., a widthset by the LPI of the lens sheet or lens array), and the printer used toprovide the printed interlaced image may have a particular resolution(e.g., dots per inch (DPI)). Hence, these limitations or parameters of alenticular product or assembly (e.g., a security stamp or securitythread for a bank note or piece of currency) define the maximum numberof frames or images that can be interlaced and then printed on a lenssheet by the equation: Maximum number of frames=DPI/LPI.

FIG. 1 illustrates a cross-sectional view (or end view) of a very simplelenticular device or assembly 100 that is useful for discussing theselimitations associated with traditional lenticular printing andinterlacing. As shown, the assembly 100 includes a single lenticule orelongated lens 110 with a planar side or base 112 of a particular width,L_(W) (lenticule size as defined by the LPI of a sheet includinglens/lenticule 110). An ink layer or printed interlaced image 120 isprovided directly onto the back side or base 112 of the lenticule 110,and, in this example, the interlaced image 120 includes five imageslices 124 (e.g., long, thin portions of five different images/frames)that would extend the length of the lenticule 110 in a parallel manner(parallel to each other and to the longitudinal axis of thelenticule/lens 110).

In the assembly or device 100, the lens size, L_(W), and pixel size issuch that the lens 110 can only work well with a maximum of fiveinterlaces or image slices 124 (e.g., five pixels with each pixel beingassociated with one of the five interlaced frames/images). These areshown to be exactly aligned with the lens 110 but may, in practice, besomewhat offset while still being parallel to the longitudinal axis ofthe lens 110 and still achieve a desirable image when viewed through thelens 110. The interlacing is orthogonal in that the five pixels extendorthogonally across the lens 110 relative to its longitudinal axes(e.g., the elongated slices of the image extend parallel to thelongitudinal axis of the lens 110 such that side-by-side pixelsassociated with these slices/interlaces extend across the lens width,L_(W)).

However, in order to achieve a 3D effect with lenticular sheets, theminimum number of frames needed is six or more images/frames. Thismeans, for example, that for a 1200 DPI output device (e.g., printer)the lenticular lenses must have a width associated with a 200 LPI orhigher (where LPI=DPI/Number of frames or, in this case, 200 LPI=1200DPI/6 frames). This relationship between resolution of the outputdevice, the number of frames needed to produce 3D, and the lens sizecreates a significant restriction to developing thinner lenticules andcorresponding thinner lenticular products (such as security threads orstamps for currency or bank notes). However, it should be understoodthat the limitation is not the ability to fabricate thinner lens sheetsbecause lens sheets that are very thin can readily be produced withpresently available technology. Instead, the restriction or challenge toproviding thin lens sheets comes from the high resolution that would berequired, and, therefore, the limitation of the number of frames thatcan be printed on or underneath smaller sized lenses (e.g., lenses withsmaller widths or L_(W)).

FIG. 2 illustrates a top perspective view of a lenticular product orassembly 200 that may use conventional or traditional interlacing. Asshown, the assembly 200 includes a lens sheet or lens array 210 that maybe formed of a thickness of plastic or other transparent material. On atop or exposed side, the lens sheet 210 is grooved or shaped to providea number of lenticules or elongated lenses 214 that extend in a parallelmanner from one end to the other of the sheet 210. As is common, thelenticules 214 extend “vertically” in the array or sheet 210 or withtheir longitudinal axes being orthogonal to the top and bottom edges211, 213 of the sheet 210 (or being parallel to left and right sideedges). Each lenticule or lens 214 has a size or width, L_(W), that isdefined by the LPI of the lens sheet 210.

In the lenticular assembly 200, an ink layer 220 is printed directlyupon a planar back side or bottom side 216 of the lens sheet 210 (or maybe provided on a substrate that is laminated onto the lens sheet 210).The ink layer 220 is printed to provide a number of interlaced images orslices 224 underneath each lenticule 214 such as to provide a 3D effect.As shown, the interlaced image of ink layer 220 has five slices 224associated with five different frames underneath each lenticule 214,with different slices of the same frame being provided under differentlenticules 214 in the sheet 210. In this case, the image file forprinted ink layer 220 was created with five pixels to match the size,L_(W), of each lens 214.

Lenticular devices may also use lenses or lenticules that are providedin a sheet or array with an angular arrangement, e.g., not parallel ororthogonal to edges of the sheet/array. FIG. 3 illustrates aconventional slant lens lenticular device or assembly 300 in which alens or lenticular sheet 310 is combined with an interlaced imageprovided in an ink layer 320. The lens sheet 310 includes a number oflenticules or lenses 314 on a top or exposed side, and the lenticules314 extend parallel to each other but, in this lens sheet 310, thelenticules 314 are not arranged vertically or horizontally. In otherwords, the lenticules or lenses 314 are “slanted” with theirlongitudinal axes, Axis_(Long), as shown at 315 arranged to each be at aparticular angle, θ, relative to a side edge 311 of the lens sheet 310,with the slant angle, θ, being less than 90 degrees (not orthogonal)such as 20 to 60 degrees or the like. Again, each lens 310 has a size orwidth, L_(W), set by the LPI of the sheet 310 that may limit the numberof image slices that may be placed underneath each of the lenses 314with conventional interlacing techniques.

The lenticular assembly 300 further includes an ink layer 320 providinga printed interlaced image with a number (here five) of slices 324provided under each lens 314. In other words, instead of having theinterlaces or slices 324 provided with “vertical” strips that arespliced together, the ink layer 320 provides the image with slantedstrips 324 matching the slant angle, θ, of the lenses 314. Hence, theinterlacing for slant lens sheets such as sheet 310 has traditionallyinvolved arranging the elongated slices of a number of images to extendparallel to each other and also to the longitudinal axis, Axis_(Long),as shown at 315 of the lenses 314. Hence, the interlacing of the device300 again is to match the size of the lenses 314 with five pixelsarranged orthogonally to the longitudinal axis, Axis_(Long) (e.g., toextend across the width, L_(W), of the lens 314). As can be seen, theuse of slant lens does not increase the amount of information providedunder the lens array when traditional interlacing is used to generatethe interlaced image.

There remains a need for methods for providing an interlaced image(i.e., interlacing methods) that allow additional information to beprovided under the lenses or lenticules of a sheet of lenticularmaterial (or a lens sheet). Preferably, such interlacing methods wouldbe useful with existing and to-be-built output devices (e.g., printers)to allow lenticular products to be provided with desirable imagery(e.g., 3D imagery) with much lower thicknesses of lenticular material orlens sheets, e.g., to support use of lenticular assemblies or elementsas security threads, stamps, and the like in bank notes, currency, andother items.

SUMMARY

The inventors recognized that thinner lenses and, therefore, thinnerlenticular material could be used to display 3D and other imagery byprinting more information under each lens or lenticule. To this end, aninterlacing method was developed for use with angled lenticules or slantlenses that differs from traditional interlacing, in part, by utilizingnon-orthogonal interlacing.

Interlacing of images into a combined image file (or interlaced printfile for controlling an output device or printer) involves arranging aset of pixels in a line or column that is transverse but non-orthogonalto the longitudinal axis of a slant lens or slanted lenticule. Each ofthese pixels is associated with a different frame/image, e.g., 6 or moreframes are typically used in each interlaced image, with one beingvisible through the lens or lenticule at a time by a viewer. The newslant lens interlacing method does not involve slicing each frame andthen splicing these slices together. Instead, individual pixels fromeach frame are combined within a digital print file in a unique patternto provide the non-orthogonal interlacing described herein (e.g., thenew interlacing may be considered “matrix interlacing” or“angular-offset interlacing”).

By utilizing non-orthogonal interlacing or matrix interlacing to createa print file that is used to control an output device/printer, asignificantly larger amount of information may be presented under eachslant lens or slanted lenticule. For example, the traditionalinterlacing example provided in FIG. 3 was able to provide five pixelsunder each lenticule while the new interlacing process is able toprovide 10 to 14 pixels under the same-sized lenticule.

More particularly, a method is provided for generating an interlacedprint file for use in printing on or for a lens sheet with slantedlenticules. The method includes providing digital files for a set offrames for interlacing and inputting a slant angle for lenticules of thelens sheet. The method further includes interlacing the frames to forman interlaced print file by positioning a number of pixels from each ofthe frames in the print file based on a pattern of pixel locationsprovided in a predefined pixel matrix. The pixel matrix is configured tomap the plurality of pixels to the lenticules of the lens sheet based onthe slant angle. To this end, interlacing of the frames isnon-orthogonal to a longitudinal axis of each of the lenticules (i.e.,not directly across the width of each lens as in traditional slant lensinterlacing).

In some implementations of the method, each set of interlaced ones ofthe pixels associated with the set of frames is aligned in a column orin a row of the pixel matrix (e.g., interlacing is along a verticalline/column (or horizontal line/row) while the lenses are slanted fromvertical (or from horizontal)). The pixel matrix may be made up of anumber of spaced apart sets of the pixels from each of the frames withthe pixel locations for each of the spaced apart sets being arrangedlinearly at an offset angle of about the slant angle. In such cases, theslant angle may be in the range of 14 to 15 degrees, in the range of 18to 19 degrees, in the range of 26 to 27 degrees, or in the range of 44to 46 degrees. The number of frames in the set of frames may vary withsome embodiments interlacing 6 frames, 9 frames, or 16 frames to createa desired effect (e.g., 3D imagery visible through a lens sheet). Inthese cases, the pixel matrix comprises a repeating 6 by 6 pixel matrix,a 9 by 9 pixel matrix, or a 16 by 16 pixel matrix.

According to another aspect of the present description, a lenticularproduct is provided that includes a transparent lens sheet and an imageor ink layer. The lens sheet includes a first side having a plurality ofside-by-side, elongated lenses each at an angle in the range of 10 to 46degrees from a vertical or a horizontal axis of the lens sheet. The inklayer is proximate to a second side of the lens sheet opposite the firstside (e.g., is printed onto this planar side or is laminated to the sidewith transparent adhesive). The ink layer includes pixels from aplurality of digital images, with the pixels being arranged in a patternof pixel locations providing non-orthogonal interlacing of the digitalimages relative to each of the elongated lenses. The pattern of pixellocations can be adapted to align a number of the pixels from each ofthe digital images to be parallel to a longitudinal axis of an adjacentone of the elongated lenses such that pixels of only one digital imageare viewed at a time through the lens sheet's lenses.

In some cases, the lens sheet has a thickness in the range of 10 to 2500microns and the elongated lenses are provided on the first side at 75 to1500 LPI (which defines the width or size of each lens). The pluralityof digital images may include a number of images selected from the rangeof 6 images to 16 images, and, further, the non-orthogonal interlacingmay be provided by sets of the pixels equal in number to the number ofimages including at least one pixel from each of the plurality ofdigital images. It may be useful for the sets of the pixels that areproviding the non-orthogonal interlacing to be aligned in a row or in acolumn of the pattern of pixel locations.

In particular implementations of the lenticular product, the angle ofthe elongated lenses is 14.04 degrees, 18.435 degrees, 26.57 degrees, or45 degrees. The plurality of digital images can be selected such thatthe plurality of pixels in the ink layer produces a 3D image when viewedthrough the elongated lenses of the lens sheet. A transparent adhesivemay be provided to attach the ink layer that is printed on a substrateonto the lens sheet or to attach the lens sheet and ink layer to asubstrate. The lenticular product may be nearly any object such as apiece of paper or polymer currency (or bank note) with a security threador stamp (e.g., a 3D image is visible with the security thread or stampusing slant lenses combined with the interlacing taught herein).

According to yet another aspect of the present description, a method isprovided for fabricating a lenticular product. The method includesproviding a pixel matrix defining a plurality of pixel locations forpixels of a number of images. The pixel locations are adapted toposition the pixels associated with each of the images in a lineparallel to longitudinal axes of lenticules provided at a slant angle ina lens sheet. The pixel locations of the matrix are further adapted tointerlace sets of the pixels associated with differing ones of theimages along an interlace line that is transverse to and non-orthogonalto the longitudinal axes of the lenticules.

The method further includes generating a digital print file byinterlacing the images according to the pixel matrix by placing thepixels of the images into the pixel locations identified in the pixelmatrix. Then, with the digital print file, the method includes operatingan output device to print an interlaced image onto a planar back side ofthe lens sheet or onto a substrate for later application to the backside of the lens sheet. In some preferred embodiments, the slant angleis 14.04 degrees, 18.435 degrees, 26.57 degrees, or 45 degrees. In theseand other embodiments, the lenticules can be provided on the lens sheetat an LPI of at least 75 LPI, and the lens sheet may have a thickness of10 to 2500 microns.

According to a further aspect of the present description, an opticalproduct or assembly is provided that includes a transparent lens sheet.This sheet or lens material has a first side with a plurality ofside-by-side sets of linearly arranged lenses. Each set is at a slantangle in the range of 10 to 46 degrees from a vertical or a horizontalaxis of the lens sheet. The product or assembly further includes an inkor image layer, proximate to a second side of the lens sheet oppositethe first side, that includes pixels from a plurality of digital imagesor frames. The pixels are arranged in a pattern of pixel locationsproviding non-orthogonal interlacing of the digital images relative toeach of the sets of the linearly arranged lenses. The pattern of pixellocations is adapted to align a number of the pixels from each of thedigital images to be parallel to a line extending through a center ofthe linearly arranged lenses in an adjacent one of the sets of thelinearly arranged lenses.

In some preferred embodiments, each of the linearly arranged lenses hasa round base while in others each of the linearly arranged lenses has ahexagonal base or a square base. In some cases, the lens sheet has athickness in the range of 10 to 2500 microns and the sets of thelinearly arranged lenses are provided on the first side at 75 to 1500LPI. Further, each of the linearly arranged lenses has a size parametermatching an LPI of the sets of the linearly arranged lenses (e.g., awidth of a square or hexagonal lens or a diameter of a round lens may bechosen to match the LPI (e.g., a corresponding width of a lenticuleprovided at such an LPI)).

In many cases, the plurality of digital images includes a number ofimages selected from the range of 5 images to 16 images, and thenon-orthogonal interlacing is provided by sets of the pixels equal innumber to the number of images including at least one pixel from each ofthe plurality of digital images. In these cases, the sets of the pixelsproviding the non-orthogonal interlacing are aligned in a row or in acolumn of the pattern of pixel locations. Further, it is sometimesuseful for the slant angle to be within a range of 14 to 15 degrees, arange of 18 to 19 degrees, a range of 26 to 27 degrees, or a range of 44to 46 or more specifically for the slant angle to be 14.04 degrees,18.435 degrees, 26.57 degrees, or 45 degrees.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a cross-section of a simple lenticular device or assemblyillustrating conventional interlacing;

FIG. 2 illustrates a top perspective view of a conventional lenticulardevice or product;

FIG. 3 illustrates a top perspective view similar to FIG. 2 showing asecond conventional lenticular device or product that uses slant lenswith conventional orthogonal interlacing;

FIG. 4 is a diagram illustrating use of new non-orthogonal interlacing(matrix interlacing) to take advantage of a greater amount of printingspace under a slant lens or slanted lenticule;

FIG. 5 is a diagram of traditional interlacing and an example ofnon-orthogonal or matrix interlacing as taught herein, with both beingused with a slant lens;

FIGS. 6-9 each illustrate a diagram of an exemplary step ofnon-orthogonal interlacing that may be used in generating a pixel mapfor creating a digital print file;

FIG. 10 illustrates a pixel map or matrix of pixels arranged in apattern useful for interlacing nine frames or images to provide the18.435 degree, non-orthogonal configuration discussed with reference toFIG. 7;

FIGS. 11A-11C illustrate, schematically, end views of lenticularproducts or assemblies that may be fabricated using the non-orthogonalinterlacing or matrix interlacing taught herein;

FIG. 12 illustrates a flow diagram of a method of fabricating alenticular produce assembly combining slant lens material with an imageprinted according with the non-traditional interlacing taught herein;

FIG. 13 illustrates a functional block diagram of a system for printinga non-orthogonally interlaced image for use with slanted lenticules(e.g., with lenticular material with slanted lenses);

FIG. 14 illustrates an optical product with a lens sheet having lenseswith round-shaped bases (or round lenses) arranged in linear setsarranged at slant angles for use with an image or ink layer printedusing a pixel map or matrix of pixels (such as the pixel map of FIG.10);

FIG. 15 illustrates an optical product, similar to that shown in FIG.14, with a lens sheet having lenses with hexagonal-shaped bases (orhexagonal lenses); and

FIG. 16 illustrates an optical product, similar to that shown in FIG.14, with a lens sheet having lenses with square-shaped bases (or squarelenses).

DETAILED DESCRIPTION

Briefly, the present description is directed toward a method forlenticular interlacing for use with lens sheets or lenticular materialhaving slanted lenticules or slant lenses (slant lens interlacing,matrix interlacing, and non-orthogonal interlacing, interchangeably).The interlacing differs from traditional interlacing because it does notsimply involve providing slices of an image (or pixels associated witheach) orthogonally or directly across the width of the lens (or withslices arranged in parallel at an angle matching the lens). Instead,each frame or image is first considered as a set of pixels, and pixelsfrom each frame or image are arranged in a matrix or pattern such that aset of pixels made up of a pixel from each frame is arranged transversebut non-orthogonally under the lenticule. In this way, a much largernumber of pixels for a particular output device resolution (DPI) can beprovided under a lenticule for selective viewing. As a result, thenon-orthogonal interlacing supports use of a thinner lens sheet toachieve a particular imagery or supports a much better quality imageryto be viewed with a predefined lens sheet thickness.

In FIG. 3, the image provided in the print layer 320 withinterlaces/slices 324 is basically the same as the image provided withprint layer 220 in FIG. 2. Particularly, the interlaces/slices 324 arearranged with the same angle as the lenses 314, and the amount ofinformation or frames is also limited by the same relationship betweenthe lenses, DPI, and resolution. In order to break this relationship(i.e., frames multiplied by LPI equals DPI) that limits the amount ofinformation or, in this case, pixels that can be printed under thelenses, the inventors recognized that it would be useful to use acompletely different pixel array (or pixel map) for interlacing theimages/frames under the slant lens.

FIG. 4 provides a diagram of a single slant lens or slanted lenticule400 for which it may be desirable to generate a print file for use inprinting an interlaced image. The lenticule 400 is shown to be angled(not simply vertical or horizontal on a surface of a lens sheet) withits longitudinal axis, Axis_(Long), as shown at 405 being at an offsetor slant angle, θ, to vertical (or horizontal) as shown at 407 (e.g., aside of a lens sheet or the like). Traditional interlacing would beprovided with the slant lens 400 by arranging slices parallel to theaxis 405 of the lens 400, which would provide a number of pixelsorthogonally across the lens 400 or to fill the dimension, L_(W) (e.g.,the width or size of the lens 400 as defined by its LPI). In contrast,the non-orthogonal interlacing of the present description calls forproviding a number of interlaced pixels transverse to the longitudinalaxis 405 but not orthogonally, i.e., along the line 420 which has alength or dimension, D_(Interlacing), that is much larger than thewidth, L_(W), of the lens 400.

In a slant lens 400 as shown in FIG. 4, there is a triangle withproperties that can be used to fit more information under the lens 400.The slant lens 400 is defined by a size, L_(W), as shown by line 428,which is given by: L_(W)=1/LPI. If, for example, a lens sheet isfabricated at 75 LPI, the size, L_(W), of each lenticule or lens 400would be 1/75 inches or 0.0133 inches. However, the vertical distance,D_(Interlacing), as shown with line 420 (or the hypotenuse of thetriangle) is larger than the lens size, L_(W), and this magnitude ofthis vertical distance, D_(Interlacing), is defined or given by thetriangle that is formed and shown in FIG. 4. Specifically, the triangleformed by or made up of a segment/length of the lens 400 as shown byline 424, the width of the lens 400 as shown by line 428 (which isorthogonal to the longitudinal axis 405 of the lens 400), and thevertical distance, D_(Interlacing), as shown by line 420 contains anangle, a, (between lines 420 and 424). This triangle may be defined bythe equation: D_(Interlacing)=L_(W)/sin(a). In turn, this equation maybe rewritten as: D_(Interlacing)=(1/LPI)/sin(a).

Using specific values may be illustrative at this point in thedescription. For example, a lenticular array may be formed at 75 LPI,which provides a lens size or width, L_(W), of 0.0133 inches. If theangle, a, is taken to be 25 degrees (as one useful, but non-limitingexample), the vertical distance, D_(Interlacing), is 0.0315 inches,which is nearly three times the lens width, L_(W). Hence, one canreadily appreciate why it is desirable to provide the interlacing orinterlaced pixels along the line 420 rather than along the orthogonal orline 428. Using the larger distance, D_(Interlacing), to arrange thepixels in a vertical position with a slanted lens 400 provides much moreroom or printing space than going with the traditional interlacingacross the lens 400.

However, it was also understood by the inventors that traditionalinterlacing techniques could not be used to provide information underthe line 420 to allow viewing of a quality image such as 3D imagery with6 or more interlaced frames/images. Instead, FIG. 5 illustrates adiagram 500 comparing traditional interlacing with a new non-orthogonalor matrix interlacing process to place pixels or information under thisvertical distance or the hypotenuse of the triangle discussed withreference to FIG. 4.

As shown, a slant lens 510 is provided as may be included in a lenssheet or piece of lenticular material in a lenticular device/assembly(such as a security thread or stamp for currency or bank notes or otheritems). Traditional interlacing is shown with the set of pixels 520extending side-by-side orthogonally across the lens 510. The size,L_(W), of the lens 510 limits the number of pixels 520 with five pixelsbeing shown in this example.

In contrast, though, non-orthogonal or matrix interlacing is shown withthe set of pixels 530 extending transverse but non-orthogonally acrossthe longitudinal axis of the lens 510. Specifically, the pixel set (orinterlace set) 530 is made up of a number of side-by-side pixels 531,532, 533, 534, 535, 536, 537, 538, 539, 540, 541 from a like number offrames or images being interlaced/combined to produce imagery viewablevia lens 510. In this example, the pixels of each of the sets 520 and530 are of the same size but there is room along the vertical orhypotenuse for a greater number of such pixels (e.g., 5 pixels in thetraditional interlacing set 520 compared with 12 pixels in thenon-orthogonal interlacing set 530, which is more than a doubling of thenumber of pixels or amount of information that can be printed under orprovided under the slant lens 510). The dashed box 590 is useful forhighlighting or showing an exemplary lens focus for the lens 510, whichshows that with the new interlacing 530 (and a number of otherinterlacing sets similar to set 530) the lens 510 is still focusing onpixels 539, 549, 559, 569 that belong to or are associated with a singleframe or image.

The diagram 500 of FIG. 5 and the illustrated ideal or goal interlacing530 is useful for demonstrating that the use of non-orthogonalinterlacing would be desirable for increasing the amount of informationthat can be placed under a slant lens 510. However, a limitation facingthose skilled in the art trying to fabricate a slant lens lenticulardevice is how to work with printing limitations of output devices orprinters used to provide or print the interlaced image (e.g., providethe ink layer underneath a lens sheet). With this problem in mind, theinventors recognized that it would be desirable to provide a matrix ofpixels or a pixel map that can be used to generate a print file forcontrolling a printer/output device. In other words, each image or framemay be first stored digitally as a set of pixels, and pixels from eachframe may be arranged or combined (interlaced) according to a predefinedmatrix of pixels or pixel map to achieve a print file withnon-orthogonal interlacing suited for a particular slanted lenticularmaterial (e.g., a transparent lens sheet with lenses slanted fromvertical/horizontal at a particular or predefined angle).

FIG. 6 illustrates a diagram or schematic 600 of a step in anon-orthogonal interlacing process that may be carried out to generate apixel map for creating a digital print file from a set of frames orimages (e.g., 6 or more frames used in creating a 3D graphic under slantlens lenticular material). A first frame or image may be chosen forprocessing, and this image/frame may be pixilated or separated intoindividual pixels of a certain size and location (X, Y coordinates)within the image/frame. Then, as shown in FIG. 6, a blank pixel map 610is provided made up of rows and columns of pixels 612 (or pixellocations for receiving or being assigned pixels from the interlacedimages). Then, as shown in FIG. 6, a string or line 620 of pixels 624from a single frame/image of the set of frames/images to beinterlaced/combined are positioned in the pixel map 610 atlocations/coordinates 612 to follow a particular angle, a, which ismeasured as shown between a longitudinal axis (or edge) of a slant lensand a vertical line (which defines the interlacing distance,D_(Interlacing), for the slant lens).

Due to the rigidity of the pixel map 610 with its orthogonal rows andcolumns of pixel locations 612, the inventors understood that it ispreferable to map the pixels 624 of string/line 620 to follow apredefined angular offset. Here, angle, a, is 26.57 degrees (e.g., 20 to30 degrees) as the pixels 624 are arranged in a pattern to define a lineor string 620 (shown to be linear with dashed line 621) that will beconcurrently visible through a slant lens with a similar angular offset(e.g., from vertical or horizontal in the lens sheet).

In practice, the mapping 610 is created by placing a first pixel 624 andthen placing a next pixel at the desired angular offset (e.g.,vertically down two pixel locations 612 in the same column and over onepixel location 612 to an adjacent row when starting in an upper “left”position), and then repeating this process to the end/edge of the pixelmap 610. In this example, since sin 26.57 degrees=0.4226, theinterlacing distance, D_(Interlacing) as shown in FIG. 4 is given byD_(Interlacing)=L_(W)/sin a or L_(W)/0.4226. This provides about doublethe space to print frames (pixels associated with frames) with the samelens size, L_(W), and with the same resolution.

In FIG. 7, a blank pixel map 610 again may be provided with a pluralityof rows and columns of pixel locations (for pixels of a particular sizeto suit an output device resolution or the like). Diagram 700 shows astep of a non-orthogonal interlacing being performed to create a pixelmap for a slant lens with an angular offset of 18.435 degrees (e.g.,slant angle, a, is in the range of 15 and 20 degrees with about 18.5degrees being more ideal). As shown, pixels 724 of a single frame/imageof a set of frames/images being combined/interlaced are arranged asshown in a pixel line or/string 720 (shown to be linear with dashed line721).

Relative to vertical (or horizontal in some cases), the pixels 724 arearranged in the line/string 720 at an offset of 18.435 degrees (e.g., byplacing a first pixel 724 at a location 612 and then stepping down (orup) three locations 612 in the same column and one location 612 over ina row to an adjacent column and then repeating this process to theedge/end of the map 610). In this example, since sin 18.435degrees=0.3162, the interlacing distance, D_(Interlacing) as shown inFIG. 4 is given by D_(Interlacing)=L_(W)/sin a or L_(W)/0.3162. Thisprovides about triple the space to print frames (pixels associated withframes) with the same lens size, L_(W), and with the same resolution.

In FIG. 8, a blank pixel map 610 again may be provided with a pluralityof rows and columns of pixel locations (for pixels of a particular sizeto suit an output device resolution or the like). Diagram 800 shows astep of a non-orthogonal interlacing being performed to create a pixelmap for a slant lens with an angular offset of 14.04 degrees (e.g.,slant angle, a, is in the range of 10 and 15 degrees with about 14degrees being more ideal). As shown, pixels 824 of a single frame/imageof a set of frames/images being combined/interlaced are arranged asshown in a pixel line or/string 820 (shown to be linear with dashed line821).

Relative to vertical (or horizontal in some cases), the pixels 824 arearranged in the line/string 820 at an offset of 14.04 degrees (e.g., byplacing a first pixel 824 at a location 612 and then stepping down (orup) four locations 612 in the same column and one location 612 over in arow to an adjacent column and then repeating this process to theedge/end of the map 610). In this example, since sin 14.04degrees=0.2426, the interlacing distance, D_(Interlacing) as shown inFIG. 4 is given by D_(Interlacing)=L_(W)/sin a or L_(W)/0.2426. Thisprovides about quadruple or four times the space to print frames (pixelsassociated with frames) with the same lens size, L_(W), and with thesame resolution.

In FIG. 9, a blank pixel map 610 again may be provided with a pluralityof rows and columns of pixel locations (for pixels of a particular sizeto suit an output device resolution or the like). Diagram 900 shows astep of a non-orthogonal interlacing being performed to create a pixelmap for a slant lens with an angular offset of 45 degrees (e.g., slantangle, a, is in the range of 40 and 50 degrees with about 45 degreesbeing more ideal). As shown, pixels 924 of a single frame/image of a setof frames/images being combined/interlaced are arranged as shown in apixel line or/string 920 (shown to be linear with dashed line 921).Relative to vertical (or horizontal in some cases), the pixels 924 arearranged in the line/string 920 at an offset of 45 degrees (e.g., byplacing a first pixel 924 at a location 612 and then stepping down (orup) one location 612 in the same column and one location 612 over in arow to an adjacent column and then repeating this process to theedge/end of the map 610).

The arrangement of pixels in rows and columns presents some limitationsas to the interlacing of the pixels, but it is likely that these fourachievable slant or offset angles for use in interlacing pixels of thesame image will prove beneficial in manufacturing lenticular productswith slanted lenticules. The creation of the print file would thencontinue in each of these examples with selection of pixels of differingframes/images, and then arrangement of such pixels in a similar mannernearby to the pixels already positioned in the pixel map 610 until all(or most) of the pixel locations 612 are occupied.

FIG. 10 illustrates a pixel map or matrix of pixels 1000 arranged in apattern useful for interlacing nine frames or images to provide the18.435 degree, non-orthogonal configuration discussed with reference toFIG. 7. In this example, nine frames are to be interlaced and printed inan ink or image layer for use with a lens sheet with slanted lenticulesor slant lenses that are angled or offset from vertical by about 18.435degrees. To this end, a single lens 1010 is shown positioned over thepixel map 1000, and the longitudinal axis 1015 of the lens 1010 is shownto be at a slant or offset angle, θ, to vertical (but this could also behorizontal) 1013 of about 18.435 degrees. In the matrix 1000 thenumbered pixels or pixel locations 1002 represent locations where pixelsfrom interlaced images/frames would be located when printing isperformed using a print file built upon the map or matrix 1000.

Specifically, the matrix 1000 is used to interlace nine frames or imagesand numbers 1 through 9 are placed in each of the pixels or pixellocations 1002 in the map (e.g., each spot in the rows and columns ofthe map/matrix 1000), with each like number representing a pixel fromthe same frame/image (e.g., all pixel locations 1002 filled with a “4”would correspond to pixels from a fourth frame/image) and at locationsin such frame/image corresponding with the locations in the pixelmap/matrix 1000. For example, a “9” in the center of the map/matrix 1000corresponds with a pixel in the ninth frame/image located in about thecenter of the frame/image. As another example, a “3” located in thelower left hand corner of the map/matrix 1000 corresponds with a pixelin the lower left hand corner of the third frame/image of the set ofnine frames/images being combined to form an interlaced or combinedprint file.

The lens 1010 is useful for showing that pixels at pixel locationsunderneath the lens 1010 are aligned to be parallel with thelongitudinal axis 1015 of the lens 1010 are concurrently visible whilethe interlacing is non-orthogonal (i.e., is along a column in themap/matrix 1000 (but could be along a row if the lens 1010 were angledfrom horizontal rather than vertical)). For example, as shown at 1040, aset of “9” pixels provided at the pixel locations 1040 in the map/matrix1000 would be visible to a viewer via the lens 1010 from a particularpoint of view. In other words, following an inclination of 18.435degrees in the map/matrix 1000 (and under the lens 1010 having this sameslant to vertical 1013) all the digits are the same within pixellocations (e.g., when used to create a print file the map/matrix 1000calls for pixels from a single image to be aligned along an inclinationof 18.435 degrees).

However, the interlacing of pixels is non-orthogonal to the lenslongitudinal axis 1015 as shown with the set 1050 of interlaced pixels(or pixel locations in the map/matrix 1000) that includes pixels fromeach frame/image. The interlacing process or algorithm may be generatedbased on the teaching of FIG. 7, in this example, and it is performed toalign pixels of like frames/images with the longitudinal axis of thelens 1010 while providing non-orthogonal or matrix interlacing of pixelsof each of the frames/images across the lens 1010.

The inventors noted during the interlacing process that repeatingsubmatrices may be identified, and these may be repeated (e.g., placedside-by-side and stacked upon each other in a repetitive manner) togenerate a map or matrix 1000 of a desired size and/or shape to suit aparticular lens sheet. One exemplary interlacing submatrix is shown at1060 that can be used in providing non-orthogonal interlacing of nineframes/images to suit a lens sheet with lenticules or lenses slanted to18.435 degrees (i.e., to place pixels from each frame at inclines orangular offsets of 18.435 degrees for proper viewing through the lens1010). Similar pixel maps or matrices can readily be generated for otherlens sheets with different angular slants or offsets from vertical (suchas for 14.04 degrees, 26.57 degrees, and 45 degrees (with ranges ofabout 5 or more degrees on either side of these values)).

FIGS. 11A and 11B illustrate, schematically, end views of lenticularproducts or assemblies that may be fabricated using the non-orthogonalinterlacing or matrix interlacing taught herein. As shown, thelenticular assembly or product 1100 includes a lens sheet or piece oflenticular material 1110 with a first (or top) side or surface with aplurality of lenticules or elongated lenses 1114 that are arranged to beangled or angularly offset (“slanted”) relative to a vertical (orhorizontal) axis of the lens sheet 1110. For example, the lenticules1114 may be slanted at 10 to 45 degrees, with some embodiments usingslant angles of 14.04 degrees, 18.435 degrees, 26.57 degrees, or 45degrees for the lenticules 1114.

The product 1100 further includes an interlaced image provided byprinting an ink layer 1120 directly onto the second (or bottom) side orsurface 1118 of the lens sheet 1110. The ink layer 1120 is printedaccording to a print file or digital combined file in which a number offrames/images have been interlaced according to the teaching provided inthis description. In one example, the lenses 1114 are provided at sizesassociated with 75 LPI up to 2500 LPI, and the use of the non-orthogonalor matrix interlacing allows the thickness, t, of the lens sheet 1110 tobe thinner than traditional interlacing as more information (or pixels)can be placed under each lens 1114. For example, the thickness, t, maybe relatively thick such as about 20 mils or be very thin down to about10 microns and still provide 3D or other quality imagery with theprinted layer or ink layer 1120 (e.g., the range of thicknesses, t, isabout 10 micron to 20 mils). The lenticular product 1100 may then belaminated upon or attached to a substrate 1130 (such as a bank note orpiece of currency) via film 1135 of transparent adhesive.

FIG. 11B illustrates another lenticular product or assembly 1150 thatmay include the lens sheet 1110 with its slanted lenticules 1114combined with the interlaced image in printed or ink layer 1120. In thisassembly 1150, though, the ink 1120 is printed onto a substrate 1158,and the lens sheet 1110 and substrate 1158 are assembled with a film ofadhesive 1154, with ink layer 1120 facing the second or back side 1118of the lens sheet 1110. In other words, the interlaced image may firstbe printed in a print step or process and then later assembled with thelens sheet 1110 to provide a lenticular assembly or product 1150. Thelenticular products 1100 and 1150 may take many forms to practice thepresent description. For example, the products 1100, 1150 may take theform of security threads or stamps for use with currency or bank notes.

FIG. 11C illustrates a lenticular product or assembly 1170 that mayinclude the lens sheet 1110 with its slanted lenticules 1114 combinedwith the interlaced image in printed or ink layer 1120. In this assembly1170, the ink 1120 is printed directly onto the back surface 1118 of thelens sheet 1110. This relatively simple construction is useful for manyobjects/products 1170 such as a polymer bank note or anotherclear-to-translucent (or “transmissive”) product (e.g., with a clearlayer 1110) that includes a printed image at the back 1118 of the lenses1114 with an ink layer. As shown, the lenticular product or assembly1170 may also include an additional layer of ink 1174 that may be usedto provide an image that faces away from the lenses 1114 that can beviewed directly by an observer from the back side of the product 1170(e.g., the side without lenses/lenticules 1114), and this image may be aregular or conventional image that does not require lenses for properviewing (focusing for the viewer on the image in the ink layer 1174provided directly onto the printed ink layer 1120 or onto a clear oropaque substrate layer supporting or covering the ink layer 1174).

In practice, the process of fabricating a lenticular product or assemblymay involve first determining or knowing the resolution available toprint a specific product and a thickness that is targeted or the goalthickness for the product. Then, based on these parameters orlimitations, a “best” or useful option is chosen in terms of thespecific angle or matrix that is to be used for the non-orthogonal ornon-traditional slant lens interlacing. Next, the mechanical or real LPIlens is defined that will be produced and used in the product to matchthese product characteristics.

FIG. 12 illustrates a method 1200 for fabricating a lenticular productsuch as a security thread or stamp for currency or bank notes using thenon-orthogonal printing of the present description. The method 1200starts at 1205 such as with designing a desired image (e.g., 3Dimagery), selecting or defining operating parameters of output device(e.g., resolution of a digital printer), selecting materials for the inklayer, lens sheet, and an adhesive/substrate if used. At step 1210, themethod 1200 continues with selecting the lenticular material to be usedin the lenticular product. This may involve selecting a transparentplastic or synthetic material that has a particular thickness (e.g., 10microns and up) such as a thickness of adjacent substrate(s) as in thecase of a currency security thread or the like.

Step 1210 also involves defining or setting the size (i.e., LPI) of thelenticules on the surface of the lenticular material and also definingor setting the slant or offset angle for each lenticule. As discussedabove, it may be useful to use a slant angle of 10 to 45 degrees withangles of 14.04 degrees, 18.435 degrees, 26.57 degrees, and 45 degreesfor the lenticules being well suited to the non-orthogonal or matrixinterlacing.

At step 1220, the method 1200 involves selecting a number of frames (ordigital image files) to use in creating a visual effect with thelenticular material selected in step 1210. To provide 3D imagery, it maybe desirable to select 6 to 12 or more frames, and the number of framesmay be selected (or limited) by the size of the pixels achievable withthe output device (DPI of the printer selected in step 1205). Note,steps 1205, 1210, and 1220 may be performed in a fully or partiallyconcurrent matter due to the interrelationships between theparameters/characteristics of a lenticular product (e.g., LPI, DPI, lenssize, and angular offset) as discussed in detail above.

At step 1230, the method 1200 continues with generating a matrix ofpixels (or pixel map) for use in interlacing the selected frames fromstep 1220. This pixel matrix may take the form of matrix 1000 (e.g., ifthe number of frames is nine and the slant angle is 18.435 degrees) ormay be generated following the processes described with reference toFIGS. 5-9 to suit a particular slant angle and number of frames (as wellas other parameters such as lens size). The pixel matrix may then bestored as a digital file for use in later steps. In some cases, aplurality of pixel matrices or pixel maps may be generated for eachcombination of interlacing parameters, and the matching pixel matrix maybe retrieved from memory at step 1230 (e.g., one skilled in the art maygenerate pixel matrices to suit each lens sheet they may use in futurefabrication processes as well as pixel matrices suited to differingnumbers of frames, lens sizes, and output device resolutions).

At step 1240, the method 1200 continues with generating a print file forcontrolling an output device (e.g., a digital printer) to print aninterlaced image. This may involve performing non-orthogonal interlacingof the frames/images chosen in 1220 using the pixel matrix of step 1230.Each frame of step 1220 may pixilated (e.g., divided into a number ofpixels matching that of the pixel map for each frame) and then thesepixels may be plugged into pixel locations for corresponding pixels fromthe frames defined in the pixel map.

The method 1200 can then continue at 1250 by using the digital printfile from step 1240 to operate an output device to print an interlacedimage (an ink layer with pixels from each frame printed according to thepixel matrix). The printed image may be provided directly on the planar,back side of a sheet of the lenticular material selected in step 1210 orit may be printed onto a substrate. Then, in step 1260, the lenticularproduct may be completed such as by attaching the lens sheet with itsprinted ink layer to a substrate (laminate lens with interlaced image onanother object such as a security stamp onto a bank note). In othercases, step 1260 may involve attaching a substrate upon which theinterlaced image was printed onto the back of a lens sheet/lenticularelement with a transparent (or at least highly translucent) adhesive.The method 1200 may then end at 1290 or may continue at step 1210 (e.g.,selecting a different lenticular material such as with lenticules at adifferent slant angle or with lenticules of a different size or a sheetwith a different thickness) or step 1220 (e.g., selecting a differentset of frames or different number of frames to create a lenticularproduct).

FIG. 13 illustrates a functional block diagram of a system 1300 usefulfor printing an image onto a substrate or lens sheet with an interlacingpattern as described herein. The system 1300 includes a controller 1310which may take the form of nearly any computing device speciallyconfigured as shown. The controller 1310 includes a processor 1312executing computer programs or readable code to perform the functions ofan interlacing module 1320. The controller 1310 also controls or managesone or more input/output devices 1314 such as a keyboard, a mouse, atouch pad and/or touch screen, a monitor, and a user interface providedgraphically on the monitor to allow an operator to interact with thecontroller (e.g., initiate the interlacing module, provide input such asto select a lenticular material with its slanted lenticules, selectframes for interlacing under the lenticular material, and so on). TheCPU 1312 also manages operation of memory 1330 which may store the codefor module 1320 in readable format.

In the memory 1330, a set of lenticular material data 1332 is storedthat defines parameters or characteristics of a lens sheet upon which aninterlaced image will be printed. For example, the data 1332 may includethe lens size (e.g., LPI used to form the lens sheet), the thickness(e.g., 10 to 2500 microns or the like), and the slant or offset anglesof the lenticules of the lens sheet. The memory 1330 also stores anumber of frames or images in digital form that are to be interlacedwith the interlacing module 1320, and these images/frames 1340 may beselected from a larger set (not shown) by the user of the controller viauser input with I/O 1314. Each of the frames/images is digital and ismade up of a number of pixels (which may be selected to have a number orresolution similar to the pixel map 1350 or a subset of the pixels 1345may be used in the interlaced or print file 1360).

The memory 1330 further is used to store a pixel matrix 1350 generatedby the interlacing module 1320, and the pixel matrix 1350 may take theform of matrix 1000 of FIG. 10 and may be generated by the interlacingmodule 1320 as discussed with reference to any of FIGS. 5-10 and 12. Theinterlacing module 1320 may further operate to generate a print file1360 from the frames/images 1340 and the pixel matrix 1350, e.g., byselecting pixels 1345 from each image and placing them in correspondingpixel locations in the pixel matrix 1350 (which is selected to suit theslant angel 1338, the number of frames 1340, and the lens size 1334).

As shown, the system 1300 further includes an output device 1380 such asa printer with a particular DPI resolution (or multiple resolutions).The controller 1310 acts to transmit control signals 1370 based on theprint file 1360 to the output device 1380. Input 1382 is provided to thecontroller 1380 in the form of a substrate or a lens sheet 1384, and theoutput device 1380 prints ink onto the substrate or planar side of thelens sheet 1384 in the pattern defined by the control signals 1370(e.g., the print file 1360). Upon completion of printing, the outputdevice 1380 outputs 1386 a product/assembly 1390 made up of thesubstrate or lens sheet 1384 and an ink layer 1394 providing the printedinterlaced image.

From the above description, it will be understood that for many yearslenticular optics have been used with interlaced printed images or as aprint medium. The general costs are high relative to normal printingbecause of the expense of the material. In addition, making thinnerlenses work with limited resolution in a digital device or withtraditional plate setting equipment makes it very difficult if notimpossible to print interlaced images on very thin lens arrays or lenssheets because it is not supported by the traditional interlacingmathematics and corresponding lens arrays.

In contrast, the present invention and description combines an angledlens or lenticule with stair-stepped interlacing or display of images(e.g., see the interlacing of FIGS. 5-10) to allow two to four times theamount of data to be printed under the slant/angled lens or lenticulewhen compared with traditional interlacing. Conversely, thenon-traditional (or non-orthogonal) interlacing taught herein supports areduction in lens sheet thickness of up to or more than two thirds (upto 67 percent or more thickness reduction) to achieve the same imageryprovided by a much thicker lens sheet or array with traditional slantlens interlacing. Hence, more than half and up to two thirds of the costof producing traditional lens arrays can be eliminated.

The non-orthogonal or matrix interlacing for slant lens taught hereinteaches that when a lens sheet is formed with lenses made at set orpredefined angles the corresponding pixels should be placed under thelenses in a grid format (e.g., see FIG. 10 for one useful pixel mappingfor 9 frames under lenses at a slant angle of 18.435 degrees (or 15 to20 degrees)). The grid format or pixel matrix is designed so that thelike pixels or pixels of a single frame are aligned with the lens orparallel to its longitudinal axis. In this way, one can use up to twothirds (or 67 percent) less print resolution to accomplish the samegraphic or can take up to two thirds (or 67 percent) of the mass out ofthe lens array and accomplish the same imagery.

For example, a digital web lens at only 5 mils can be made to print onan HP® Indigo output device at about 812 DPI using nine frames with amechanical LPI of about 270 LPI. The lens will each focus at 5 mils butprinting this web lens with traditional slant lens interlacing isimpossible with nine frames (which is a useful number of frames for 3Dimagery). Normally, with traditional interlacing, the DPI necessary toprint this lens would be LPI multiplied by frame count or, in this case,270 LPI multiplied by 9 frames or 2430 DPI. In contrast, thenon-orthogonal or matrix interlacing taught herein matched with theangle of the slant lens can be used to support a print resolution thatis about one third that required with traditional interlacing or, inthis case, a resolution of 810 DPI is useful (which is less than theresolution provided by existing output devices of 812 DPI). In otherwords, the resolution of the printer or output device can be matchednearly exactly using the non-orthogonal or matrix interlacing describedabove.

As taught, slant lenses can be engraved and indexed or made with aslight offset (e.g., like a screw) between 80 LPI and 1500 LPI, forexample, at angles between about 10 and 46 degrees. An interlaced printfile can be generated from a pixel matrix or map adapted for the slantangle of the lenses or lenticules in the lens sheet to providenon-orthogonal interlacing of pixels of differing frames as well as tocorrespond with the lens size (set by LPI) and number of frames/imagesto be interlaced. The description further teaches how to form or printan interlaced image with a number of pixels or amount of data that is atleast double that achieved with traditional interlacing for slant lens.For example, a much larger number of frames or pixels associated withsuch frames may be printed non-orthogonally (e.g., along a vertical orcolumn rather than orthogonally to the longitudinal axis of the slantlens as in traditional interlacing), with some embodiments using 6, 9,or 16 images/frames to produce an interlaced image using a pixel matrixor map. The interlaced image printed according to this descriptionresults in a lower DPI, by using the step-wise or stairway effect ofinterlacing, than with traditional slant lens interlacing formulas(e.g., DPI=LPI×Frame Count).

At this point, it may be useful to list some anticipated results thatare achievable with the non-orthogonal or matrix/grid interlacingtechniques. A lenticular product or assembly may be formed using a lenssheet with lenticules slanted at a slant angle of 14.04 degrees (suchthat the sine value is 0.2426). An interlaced image may be providedusing 16 frames or images, such that the submatrix that is repeated inthe pixel matrix or map is 16 by 16 pixels in size (e.g., see FIG. 10where a 9 by 9 pixel submatrix 1060 is repeated). In this case, the stepratio (SR) or the increase in the amount of interlaced data whencompared with traditional interlacing is 4.122. If the mechanical (oractual) LPI of the lens sheet is 77.5 LPI (or a lens chord width or sizeof 0.012903 inches), the effective LPI for interlacing (mechanical LPIdivided by SR) is 18.801553 and, as a result, the DPI is 300.824 (with adot size of 0.003324 inches) with the vertical interlacing distance,D_(Interlacing), of 0.05319 inches (as determined by (1/mechanicalLPI)/sin 14.04 degrees).

In another example, a lenticular product or assembly may be formed usinga lens sheet with lenticules slanted at a slant angle of 14.04 degrees(such that the sine value is 0.2426). An interlaced image may beprovided using 16 frames or images, such that the submatrix that isrepeated in the pixel matrix or map is 16 by 16 pixels in size (e.g.,see FIG. 10 where a 9 by 9 pixel submatrix 1060 is repeated). In thiscase, the SR is again 4.122. If the mechanical (or actual) LPI of thelens sheet is now 400 LPI (or a lens chord width or size of 0.0025inches), the effective LPI for interlacing is 97.040272 and, as aresult, the DPI is 1552.640 (with a dot size of 0.000644 inches) withthe vertical interlacing distance, D_(Interlacing), of 0.01031 inches(as determined by (1/mechanical LPI)/sin 14.04 degrees).

In another example, a lenticular product or assembly may be formed usinga lens sheet with lenticules slanted at a slant angle of 14.04 degrees(such that the sine value is 0.2426). An interlaced image may beprovided using 16 frames or images, such that the submatrix that isrepeated in the pixel matrix or map is 16 by 16 pixels in size (e.g.,see FIG. 10 where a 9 by 9 pixel submatrix 1060 is repeated). In thiscase, the SR is again 4.122. If the mechanical (or actual) LPI of thelens sheet is now 654.5 LPI (or a lens chord width or size of 0.001528inches), the effective LPI for interlacing is 158.782145 and, as aresult, the DPI is 2540.507 (with a dot size of 0.000394 inches) withthe vertical interlacing distance, D_(Interlacing), of 0.00630 inches(as determined by (1/mechanical LPI)/sin 14.04 degrees).

In still another example, a lenticular product or assembly may be formedusing a lens sheet with lenticules slanted at a slant angle of 14.04degrees (such that the sine value is 0.2426). An interlaced image may beprovided using 16 frames or images, such that the submatrix that isrepeated in the pixel matrix or map is 16 by 16 pixels in size (e.g.,see FIG. 10 where a 9 by 9 pixel submatrix 1060 is repeated). In thiscase, the SR is again 4.122. If the mechanical (or actual) LPI of thelens sheet is now 619.51 LPI (or a lens chord width or size of 0.001614inches), the effective LPI for interlacing is 150.293547 and, as aresult, the DPI is 2404.690 (with a dot size of 0.000416 inches) withthe vertical interlacing distance, D_(Interlacing), of 0.00665 inches(as determined by (1/mechanical LPI)/sin 14.04 degrees).

In other cases, a lenticular product or assembly may be formed using alens sheet with lenticules slanted at a slant angle of 18.435 degrees(such that the sine value is 0.3162). An interlaced image may beprovided using 9 frames or images, such that the submatrix that isrepeated in the pixel matrix or map is 9 by 9 pixels in size (e.g., seeFIG. 10 where a 9 by 9 pixel submatrix 1060 is repeated). In this case,the step ratio (SR) or the increase in the amount of interlaced datawhen compared with traditional interlacing is 3.16260. If the mechanical(or actual) LPI of the lens sheet is 210 LPI (or a lens chord width orsize of 0.004762 inches), the effective LPI for interlacing (mechanicalLPI divided by SR) is 66.401062 and, as a result, the DPI is 597.618(with a dot size of 0.001673 inches) with the vertical interlacingdistance, D_(Interlacing), of 0.01506 inches (as determined by(1/mechanical LPI)/sin 18.435 degrees).

In another example, a lenticular product or assembly may be formed usinga lens sheet with lenticules slanted at a slant angle of 18.435 degrees(such that the sine value is 0.3162). An interlaced image may beprovided using 9 frames or images, such that the submatrix that isrepeated in the pixel matrix or map is 9 by 9 pixels in size (e.g., seeFIG. 10 where a 9 by 9 pixel submatrix 1060 is repeated). In this case,the step ratio (SR) or the increase in the amount of interlaced datawhen compared with traditional interlacing is 3.16260. If the mechanical(or actual) LPI of the lens sheet is 285.71 LPI (or a lens chord widthor size of 0.003500 inches), the effective LPI for interlacing(mechanical LPI divided by SR) is 90.340226 and, as a result, the DPI is813.074 (with a dot size of 0.001230 inches) with the verticalinterlacing distance, D_(Interlacing), of 0.01107 inches (as determinedby (1/mechanical LPI)/sin 18.435 degrees).

In a similar example, a lenticular product or assembly may be formedusing a lens sheet with lenticules slanted at a slant angle of 18.435degrees (such that the sine value is 0.3162). An interlaced image may beprovided using 9 frames or images, such that the submatrix that isrepeated in the pixel matrix or map is 9 by 9 pixels in size (e.g., seeFIG. 10 where a 9 by 9 pixel submatrix 1060 is repeated). In this case,the step ratio (SR) or the increase in the amount of interlaced datawhen compared with traditional interlacing is 3.16260. If the mechanical(or actual) LPI of the lens sheet is 446.28 LPI (or a lens chord widthor size of 0.002241 inches), the effective LPI for interlacing(mechanical LPI divided by SR) is 141.111744 and, as a result, the DPIis 1270.024 (with a dot size of 0.000787 inches) with the verticalinterlacing distance, D_(Interlacing), of 0.00709 inches (as determinedby (1/mechanical LPI)/sin 18.435 degrees).

In another similar example, a lenticular product or assembly may beformed using a lens sheet with lenticules slanted at a slant angle of18.435 degrees (such that the sine value is 0.3162). An interlaced imagemay be provided using 9 frames or images, such that the submatrix thatis repeated in the pixel matrix or map is 9 by 9 pixels in size (e.g.,see FIG. 10 where a 9 by 9 pixel submatrix 1060 is repeated). In thiscase, the step ratio (SR) or the increase in the amount of interlaceddata when compared with traditional interlacing is 3.16260. If themechanical (or actual) LPI of the lens sheet is 252 LPI (or a lens chordwidth or size of 0.003968 inches), the effective LPI for interlacing(mechanical LPI divided by SR) is 79.681275 and, as a result, the DPI is717.142 (with a dot size of 0.001394 inches) with the verticalinterlacing distance, D_(Interlacing), of 0.01255 inches (as determinedby (1/mechanical LPI)/sin 18.435 degrees).

In yet another similar example, a lenticular product or assembly may beformed using a lens sheet with lenticules slanted at a slant angle of18.435 degrees (such that the sine value is 0.3162). An interlaced imagemay be provided using 9 frames or images, such that the submatrix thatis repeated in the pixel matrix or map is 9 by 9 pixels in size (e.g.,see FIG. 10 where a 9 by 9 pixel submatrix 1060 is repeated). In thiscase, the step ratio (SR) or the increase in the amount of interlaceddata when compared with traditional interlacing is 3.16260. If themechanical (or actual) LPI of the lens sheet is 845 LPI (or a lens chordwidth or size of 0.001183 inches), the effective LPI for interlacing(mechanical LPI divided by SR) is 267.185227 and, as a result, the DPIis 2404.701 (with a dot size of 0.000416 inches) with the verticalinterlacing distance, D_(Interlacing), of 0.00374 inches (as determinedby (1/mechanical LPI)/sin 18.435 degrees).

In other cases, a lenticular product or assembly may be formed using alens sheet with lenticules slanted at a slant angle of 26.57 degrees(such that the sine value is 0.4473). An interlaced image may beprovided using 6 frames or images, such that the submatrix that isrepeated in the pixel matrix or map is 6 by 6 pixels in size (e.g., seeFIG. 10 where a 9 by 9 pixel submatrix 1060 is repeated). In this case,the step ratio (SR) or the increase in the amount of interlaced datawhen compared with traditional interlacing is 2.23560. If the mechanical(or actual) LPI of the lens sheet is 111.7 LPI (or a lens chord width orsize of 0.008953 inches), the effective LPI for interlacing (mechanicalLPI divided by SR) is 49.964215 and, as a result, the DPI is 299.780(with a dot size of 0.003336 inches) with the vertical interlacingdistance, D_(Interlacing), of 0.02001 inches (as determined by(1/mechanical LPI)/sin 26.57 degrees).

In a similar example, a lenticular product or assembly may be formedusing a lens sheet with lenticules slanted at a slant angle of 26.57degrees (such that the sine value is 0.4473). An interlaced image may beprovided using 6 frames or images, such that the submatrix that isrepeated in the pixel matrix or map is 6 by 6 pixels in size (e.g., seeFIG. 10 where a 9 by 9 pixel submatrix 1060 is repeated). In this case,the step ratio (SR) or the increase in the amount of interlaced datawhen compared with traditional interlacing is 2.23560. If the mechanical(or actual) LPI of the lens sheet is 223.5 LPI (or a lens chord width orsize of 0.004474 inches), the effective LPI for interlacing (mechanicalLPI divided by SR) is 99.973162 and, as a result, the DPI is 599.829(with a dot size of 0.001667 inches) with the vertical interlacingdistance, D_(Interlacing), of 0.01000 inches (as determined by(1/mechanical LPI)/sin 26.57 degrees).

In another similar example, a lenticular product or assembly may beformed using a lens sheet with lenticules slanted at a slant angle of26.57 degrees (such that the sine value is 0.4473). An interlaced imagemay be provided using 6 frames or images, such that the submatrix thatis repeated in the pixel matrix or map is 6 by 6 pixels in size (e.g.,see FIG. 10 where a 9 by 9 pixel submatrix 1060 is repeated). In thiscase, the step ratio (SR) or the increase in the amount of interlaceddata when compared with traditional interlacing is 2.23560. If themechanical (or actual) LPI of the lens sheet is 473.2 LPI (or a lenschord width or size of 0.002113 inches), the effective LPI forinterlacing (mechanical LPI divided by SR) is 211.665772 and, as aresult, the DPI is 1269.974 (with a dot size of 0.000787 inches) withthe vertical interlacing distance, D_(Interlacing), of 0.00472 inches(as determined by (1/mechanical LPI)/sin 26.57 degrees).

In yet another similar example, a lenticular product or assembly may beformed using a lens sheet with lenticules slanted at a slant angle of26.57 degrees (such that the sine value is 0.4473). An interlaced imagemay be provided using 6 frames or images, such that the submatrix thatis repeated in the pixel matrix or map is 6 by 6 pixels in size (e.g.,see FIG. 10 where a 9 by 9 pixel submatrix 1060 is repeated). In thiscase, the step ratio (SR) or the increase in the amount of interlaceddata when compared with traditional interlacing is 2.23560. If themechanical (or actual) LPI of the lens sheet is 894 LPI (or a lens chordwidth or size of 0.001119 inches), the effective LPI for interlacing(mechanical LPI divided by SR) is 399.892646 and, as a result, the DPIis 2399.317 (with a dot size of 0.000417 inches) with the verticalinterlacing distance, D_(Interlacing), of 0.00250 inches (as determinedby (1/mechanical LPI)/sin 26.57 degrees).

Although the invention has been described and illustrated with a certaindegree of particularity, it is understood that the present disclosurehas been made only by way of example, and that numerous changes in thecombination and arrangement of parts can be resorted to by those skilledin the art without departing from the spirit and scope of the invention,as hereinafter claimed.

The matrix for a 45 degree offset or slant angle typically would be 5 by5 frames (or 5×5 pixels). Note, also, there are some cases where usingthe techniques described herein allows one to gain more space than theminimum requirement. For example, when a multiple of each matrix is usedto generate the pixel matrix or map, the overall pixel matrix or mapwould be a multiple of the base or submatrix, e.g., a 9 by 9 pixelsubmatrix may be repeated in a 18 by 18 pixel matrix or pixel map (whichis a multiple of the 9 by 9 repeated pattern or submatrix).

In FIGS. 4, 5, 10, 11A, and 11B, the lens sheets were all shown anddescribed as including linear (or elongated) lenses or lenticules thatwere arranged to be slanted (e.g., not orthogonal to an edge of the lenssheet). The inventors recognized that there are many applications whereit is desirable to use lenses (e.g., microlenses when trying to achievea very thin lens sheet) that are not linear or are not lenticules.

While other interlacing techniques may be used with such lenses, it wasdetermined through analysis and experimentation that the pixel matricesor maps described above could be effectively used to provide or printinterlaced images that can then be viewed through non-linear lenses (notlenticules). However, the non-linear lenses or microlenses have to bearranged in a specific pattern to provide proper viewing of the pixelsin a manner similar to that achieve with slanted lenticules.Specifically, lens sheets are designed and produced in which thenon-linear lenses are arranged in a plurality of side-by-side (andparallel) sets of lenses, with each set of lenses being slanted on theexterior surface of the lens sheet (or lens material).

In other words, a line passing through a center point of each lens in alens set is at an angle relative to a side or edge of the lens sheet orlens material. This line is similar to a longitudinal axis of one of thelenticules described above, and these lines passing through differentsets of lenses are parallel to each other. As with the lenticules, theslant angle would fall within a range of 10 to 46 degrees from avertical or a horizontal axis (or side or edge) of the lens sheet. Thepixel maps or matrices shown in FIGS. 6-9 may be used with these slantedsets of lenses, and, in such cases, the slant angle may be in the rangeof 14 to 15 degrees, in the range of 18 to 19 degrees, in the range of26 to 27 degrees, and in the range of 44 to 46 degrees, respectively. Inparticular implementations of the optical product (when sets ofnon-linear lenses are used the product would be labeled as an opticalproduct rather than a lenticular product), the slant angle of the lenssets is 14.04 degrees, 18.435 degrees, 26.57 degrees, or 45 degrees.

Lens sheets manufactured with slanted lens sets would be used to producethe optical products such as those shown at 1100, 1150 in FIGS. 11A and11B with the lenticular material, lens sheet, or lens film 1110 replacedwith lens sheet(s)/film(s) with slanted lens sets. Similarly, the method1200 of manufacturing a lenticular product would be modified to produceoptical products by changing step 1210 to select lens sheets or lensmaterial with slanted sets of non-linear lenses (e.g., replace“lenticules” with slanted set of lenses). Additionally, the system 1300of FIG. 13 can readily be modified to produce optical products 1390 byreplacing the lens sheet 1384 with lenticules with one formed withslanted sets of lenses.

FIG. 14 illustrates, in a schematic manner, an optical product 1400 thatimplements these concepts. As shown, the optical product 1400 includeslens sheet or material (e.g., a transparent-to-transmissive (ortranslucent) film) 1410 with an upper surface 1414 that is fabricated toinclude a plurality of round-based, non-linear lenses or microlenses1422 (or round lenses) in place of lenticules. The product 1400 alsoincludes an ink layer underneath the lens sheet 1410, and the ink layeris printed using a pixel map or matrix of pixels as described for map1000 of FIG. 10. Particularly, the pixels such as pixels 1456 arearranged in 9 by 9 matrices as shown with matrix 1450. In this way, thepixels are arranged in a pattern useful for interlacing nine frames orimages to provide the 18.435 degree, non-orthogonal configurationdiscussed with reference to FIG. 7. In this example, nine frames are tobe interlaced and printed in an ink or image layer for use with the lenssheet 1410 with slanted sets of lenses or lens sets 1420, 1430, 1434,1438 that are angled or offset from vertical by about 18.435 degrees(i.e., θ equals 18.435 degrees in this example).

To this end, lens set 1420 is shown positioned over the ink layer formedbased on the pixel map (e.g., pixel map 1000 of FIG. 10. A singlelenticule or elongated lens 1440 is shown over the top of the lens set1420. This is useful for showing that the set 1420 of round lenses 1422can be used to replace the lenticule 1440. The longitudinal axis of thelenticule 1440 coincides with a line 1423 passing through the center orcenter point of each of the lenses 1422 in the lens set 1420. The line1423 is shown to be at a slant or offset angle, θ, to vertical (but thiscould also be horizontal) 1424 of about 18.435 degrees. In the matrix1450, the numbered pixels or pixel locations 1456 represent locationswhere pixels from interlaced images/frames would be located whenprinting the print layer as may be performed using a print file builtupon the map or matrix 1450.

Specifically, the matrix 1450 is used to interlace nine frames or imagesand numbers 1 through 9 are placed in each of the pixels or pixellocations 1456 in the map (e.g., each spot in the rows and columns ofthe map/matrix), with each like number representing a pixel from thesame frame/image (e.g., all pixel locations 1456 filled with a “4” wouldcorrespond to pixels from a fourth frame/image) and at locations in suchframe/image corresponding with the locations in the pixel map/matrix.For example, a “9” in the center of the map/matrix corresponds with apixel in the ninth frame/image located in about the center of theframe/image. As another example, a “3” located in the lower left handcorner of the map/matrix corresponds with a pixel in the lower left handcorner of the third frame/image of the set of nine frames/images beingcombined to form an interlaced or combined print file.

As can be seen (and as discussed with regard to FIG. 10), pixels atpixel locations underneath the lenses 1422 of set 1420 (and other sets1430, 1434, 1438) are aligned to be parallel with the line 1423 passingthrough the center of the lenses 1422 and are concurrently visible whilethe interlacing is non-orthogonal (i.e., is along a column in themap/matrix (but could be along a row if the lens set 1420 was angledfrom horizontal rather than vertical)). For example, a set of “9” pixelsprovided at the pixel locations in the map/matrix (and correspondingprinted ink layer) would be visible to a viewer via the lens set 1420from a particular point of view. In other words, following aninclination of 18.435 degrees in the map/matrix (and under the lens set1420 having this same slant to vertical 1424) all the digits are thesame within pixel locations (e.g., when used to create a print file themap/matrix calls for pixels from a single image to be aligned along aninclination of 18.435 degrees).

However, the interlacing of pixels is non-orthogonal to the lens centerline 1423 as shown with the set 1480 of interlaced pixels (or pixellocations in the map/matrix and corresponding printed ink layer) thatincludes pixels from each frame/image. The interlacing process oralgorithm may be generated based on the teaching of FIG. 7, in thisexample, and it is performed to align pixels of like frames/images withthe center line 1423 of the lens set 1420 while providing non-orthogonalor matrix interlacing of pixels of each of the frames/images across thelenses 1422 in the lens set 1420. Note, all nine pixels in the set 1480are not visible under a single lens 1422, with the example of FIG. 14showing an implementation where three lenses are covering the nineinterlaced pixels. In some cases, though, it may be useful to size andalign the lenses 1422 in some implementations of the optical product1400 so that each lens 1422 covers one pixel from each image or framebeing interlaced. In the example of FIG. 14, each lens 1422 covers or isover a set of nine pixels associated with nine different images/frames.When the printed image is viewed through the lenses, each lens 1422 inthe set is used to display a single pixel from a like image/frame (e.g.,each lens 1422 displays the “5” pixels and then the “3” pixels and soon).

As discussed above, the inventors noted during the interlacing processthat repeating submatrices may be identified, and these may be repeated(e.g., placed side-by-side and stacked upon each other in a repetitivemanner) to generate a map or matrix as shown in FIG. 14 of a desiredsize and/or shape to suit a particular lens sheet. One exemplaryinterlacing submatrix is shown at 1450 that can be used in providingnon-orthogonal interlacing of nine frames/images to suit a lens sheetwith lenticules or lenses slanted to 18.435 degrees (i.e., to placepixels from each frame at inclines or angular offsets of 18.435 degreesfor proper viewing through the lens sets 1420, 1430, 1434, 1438).Similar pixel maps or matrices can readily be generated for other lenssheets with different angular slants or offsets from vertical (such asfor 14.04 degrees, 26.57 degrees, and 45 degrees (with ranges of about 5or more degrees on either side of these values)).

In designing the lenses 1422 for use on the surface 1414 of the lenssheet 1410, it may be useful to arrange and size the lenses 1422 in eachset 1420, 1430, 1434, 1438 to replace a slanted lenticule 1440. Forexample, a particular LPI may be chosen or defined as discussed herein,and this would provide a lenticule or linear lens width, W. Then, roundlenses such as lenses 1422 may be chosen that have a lens radius that isabout one half of this lenticule width, W. The pattern of lenses 1422with such a radius is selected to provide sets 1420, 1430, 1434, 1438that are each arranged in a linear pattern such that a line passesthrough a center point of each lens 1422 in a particular set 1420, 1430,1434, 1438 as shown with line 1423. Further, such “center lines” of thesets 1420, 1430, 1434 and 1438 of linearly arranged lenses 1422 wouldall be parallel to each other. Also, as shown, there typically would belittle to no space between the lenses 1422 within a set 1420, 1430,1434, 1438 or between lenses of side-by-side sets (e.g., the lenses 1422of sets 1420 and 1430 are shown with their bases contacting each other).

The inventors further recognized that lenses with other base shapes maybe useful in optical products (in place of the slanted lenticules). Forexample, FIG. 15 shows an optical product 1500 in which the lens sheet1410 is replaced with a lens sheet 1510 with an upper surface (orsurface opposite its planar bottom surface) 1514 that is fabricated witha plurality of non-linear lenses or microlenses 1522. The lenses 1522are configured with hexagonal bases instead of round bases.

As with the lens sheet 1410, the lenses 1522 of the lens sheet 1510 arelinearly arranged (or arranged in a line or row) in a pattern such thata line 1523 passes through the center of each lens 1522 in a set 1520.Further, this center line 1523 of the set 1520 is at a slant angle, θ,and not orthogonal to the edges of the sheet 1510 (or to vertical orhorizontal axes of the sheet 1510). As shown, the slant angle, θ, isagain 18.435 degrees (but the entire range of slant angles discussedherein may be used to produce the sheet 1510). Each set 1520, 1530,1534, 1538 is arranged with their center lines parallel to each other.The lenses 1522 are used to replace or instead of the lenticule 1440,and, to this end, each of the lenses 1522 may have a width, W_(Hex) (asmeasured between opposite corners of the hexagonal base) that matchesthe lenticule width, W. Within a set such as set 1520, the lenses 1522may be arranged with abutting sides while adjacent sets may be nestedtogether with as shown (e.g., with adjacent rows staggered or offsetalong their center lines such that an external corner of an adjacentlens is received where two lenses mate in the neighboring set).

As another lens shape example, FIG. 16 shows an optical product 1600 inwhich the lens sheet 1410 is replaced with a lens sheet 1610 with anupper surface (or surface opposite its planar bottom surface) 1614 thatis fabricated with a plurality of non-linear lenses or microlenses 1622.The lenses 1622 are configured with square bases instead of round orhexagonal bases. As with the lens sheet 1410, the lenses 1622 of thelens sheet 1610 are linearly arranged (or arranged in a line or row) ina pattern such that a line 1623 passes through the center of each lens1622 in a set 1620. Further, this center line 1623 of the set 1620 is ata slant angle, θ, and not orthogonal to the edges of the sheet 1610 (orto vertical or horizontal axes of the sheet 1610). As shown, the slantangle, θ, is again 18.435 degrees (but the entire range of slant anglesdiscussed herein may be used to produce the sheet 1610).

Each set 1620, 1630, 1634, 1638 is arranged with their center linesparallel to each other. The lenses 1622 are used to replace or insteadof the lenticule 1440, and, to this end, each of the lenses 1622 mayhave a width, W_(Square) (as measured between opposite corners of thehexagonal base) that matches the lenticule width, W. Within a set suchas set 1620, the lenses 1622 may be arranged or stacked with abuttingsides (and two co-linear sides or edges) while being aligned (notstaggered from) with the lenses of adjacent or neighboring sets as shownin the example of FIG. 16. As shown at 1675, pixels from a particularframe (here Frame 6) are visible as the lenses 1622 of set 1630 arefocusing along the slant angle, θ, and this would be valid for all otherframes, too.

For any type of lens (e.g., linear or lenticule, round base, hexagonalbase, or square base) using the same mechanical LPI number (or lenticulewidth, round base diameter, hexagonal base width, or square base width),the vertical distance can be adjusted to fit the number of frames beinginterlaced (via slant interlacing as described herein). The adjustmentis performed without changing the lens size in this case. Instead, withthe same lens size, the slant angle is changed to suit the number offrames for a particular lens or lens set.

For example, the vertical distance can be increased by decreasing themagnitude of the slant angle and can be decreased by increasing themagnitude of the slant angle (e.g., more pixels can be placed in avertical column (or horizontal row if slanting is from horizontal ratherthan vertical) when a slant angle of about 14 degrees is used than whena slant angle of 45 degrees is used). In other cases, scaling can beperformed to adjust the “size” of the column of pixels or, in otherwords, the size of the matrix or map of pixels from the frames orimages. In this manner, an optical product can be designed and thenfabricated that has a lens sheet with different slant angles and/or withthe number of frames (and associated number of pixels) adjusted toprovide a desired resolution while still fitting within the verticaldistance available underneath the lenticules or lenses in each lens setfor interlacing a column of pixels.

We claim:
 1. An optical product, comprising: a transparent lens sheetcomprising a first side having a plurality of side-by-side sets oflinearly arranged lenses with each of the sets at a slant angle in arange of 10 to 46 degrees from a vertical or a horizontal axis of thelens sheet, wherein each of the sets of linearly arranged lenses has alongitudinal axis extending through a plurality of the linearly arrangedlenses and a lens width as measured by a line extending orthogonal tothe longitudinal axis across one of the lenses; and an ink layer,proximate to a second side of the lens sheet opposite the first side,comprising pixels from a plurality of digital images, wherein the pixelsare arranged in a pattern of pixel locations providing non-orthogonalinterlacing of the digital images relative to each of the sets of thelinearly arranged lenses, wherein each of the patterns of pixellocations is linear and extends along a line that is transverse andnon-orthogonal to the longitudinal axis of one of the sets of linearlyarranged lenses, the line having a length that is greater than the lenswidth.
 2. The optical product of claim 1, wherein the pattern of pixellocations is adapted to align a number of the pixels from each of thedigital images to be parallel to a line extending through a center ofthe linearly arranged lenses in an adjacent one of the sets of thelinearly arranged lenses, whereby the number of pixels is greater than anumber of pixels available using interlacing orthogonal to the lineextending though the center of the linearly arranged lenses.
 3. Theoptical product of claim 1, wherein each of the linearly arranged lenseshas a round base.
 4. The optical product of claim 1, wherein each of thelinearly arranged lenses has a hexagonal base.
 5. The optical product ofclaim 1, wherein each of the linearly arranged lenses has a square base.6. The optical product of claim 1, wherein the lens sheet has athickness in the range of 10 to 2500 microns and the sets of thelinearly arranged lenses are provided on the first side at 75 to 1500LPI.
 7. The optical product of claim 6, wherein each of the linearlyarranged lenses has a size parameter matching an LPI of the sets of thelinearly arranged lenses.
 8. The optical product of claim 1, wherein theplurality of digital images comprises a number of images selected fromthe range of 5 images to 16 images and wherein the non-orthogonalinterlacing is provided by sets of the pixels equal in number to thenumber of images including at least one pixel from each of the pluralityof digital images.
 9. The optical product of claim 8, wherein the setsof the pixels providing the non-orthogonal interlacing are aligned in arow or in a column of the pattern of pixel locations.
 10. The opticalproduct of claim 8, wherein the pattern of pixel locations comprises apixel matrix repeating submatrices of the sets of the pixels a number oftimes.
 11. The optical product of claim 1, wherein the slant angle fallswithin a range of 14 to 15 degrees, a range of 18 to 19 degrees, a rangeof 26 to 27 degrees, or a range of 44 to
 46. 12. The optical product ofclaim 11, wherein the slant angle is 14.04 degrees, 18.435 degrees,26.57 degrees, or 45 degrees.
 13. The optical product of claim 11,wherein a step ratio, defining an increase in an amount of interlaceddata when compared with traditional interlacing, is greater than 4 whenthe slant angle is in the range of 14 to 15 degrees, is greater than 3when the slant angle is in the range of 18 to 19 degrees, and is greaterthan 2 when the slant angle is in the range of 26 to 27 degrees.
 14. Theoptical product of claim 13, wherein the step ratio is 4.122 when theslant angle is in the range of 14 to 15 degrees, is 3.16260 when theslant angle is in the range of 18 to 19 degrees, and is 2.23560 when theslant angle is in the range of 26 to 27 degrees.
 15. The optical productof claim 1, further comprising an adhesive layer between the ink layerand the second side of the lens sheet, wherein the adhesive layer issubstantially transparent to light.
 16. The optical product of claim 1,further comprising a substrate and a layer of transparent adhesivebetween the substrate and the ink layer.
 17. The optical product ofclaim 1, wherein the plurality of digital images are selected such thatthe plurality of pixels in the ink layer produce a 3D image when viewedthrough the sets of the linearly arranged lenses of the lens sheet. 18.The optical product of claim 1, wherein the length of the line is atleast double the lens width.
 19. The optical product of claim 1, whereinthe length of the line is at least triple the lens width.
 20. Theoptical product of claim 1, wherein the length of the line is at leastquadruple the lens width.
 21. The optical product of claim 1, whereinthe pixels in the ink layer are static over time and wherein the pixelsconcurrently display the plurality of digital images.
 22. An assembly,comprising: a lens film comprising a plurality of side-by-side sets oflinearly arranged lenses with each of the sets at a slant angle within arange of 14 to 15 degrees, a range of 18 to 19 degrees, a range of 26 to27 degrees, or a range of 44 to 46 degrees; and an ink layer, oppositethe sets of the linearly arranged lenses, comprising individual pixelsfrom a plurality of images, whereby the pixels associated with each ofthe images is not provided in a set of strips, wherein the pixels arearranged in a pattern of pixel locations providing non-orthogonalinterlacing of the images relative to the sets of the linearly arrangedlenses, wherein the pattern of pixel locations provides sets of thepixels that each extends linearly across one of the sets of linearlyarranged lenses along a line having a length greater than a width ofeach of the sets of the linearly arranged lenses, wherein the pattern ofpixel locations is adapted to align a number of the pixels from each ofthe digital images to be parallel to a line extending through a centerof the linearly arranged lenses in an adjacent one of the sets oflinearly arranged lenses, and wherein each of the linearly arrangedlenses has a round base, a hexagonal base, or a square base.
 23. Theassembly of claim 22, wherein the lens film has a thickness in the rangeof 10 to 2500 microns and the sets of the linearly arranged lenses areprovided on the first side at 75 to 1500 LPI and wherein each of thelinearly arranged lenses has a size parameter matching an LPI of thesets of the linearly arranged lenses.
 24. The assembly of claim 22,wherein the plurality of digital images comprises a number of imagesselected from the range of 5 images to 16 images and wherein thenon-orthogonal interlacing is provided by sets of the pixels equal innumber to the number of images including at least one pixel from each ofthe plurality of digital images.
 25. The assembly of claim 24, whereinthe sets of the pixels providing the non-orthogonal interlacing arealigned in a row or in a column of the pattern of pixel locations. 26.The assembly of claim 22, wherein the slant angle is 14.04 degrees,18.435 degrees, 26.57 degrees, or 45 degrees.
 27. The assembly of claim22, wherein the plurality of digital images are selected such that theplurality of pixels in the ink layer produce a 3D image when viewedthrough the sets of the linearly arranged lenses of the lens film. 28.The assembly of claim 22, wherein a step ratio, defining an increase inan amount of interlaced data when compared with traditional interlacing,is greater than 4 when the slant angle is in the range of 14 to 15degrees, is greater than 3 when the slant angle is in the range of 18 to19 degrees, and is greater than 2 when the slant angle is in the rangeof 26 to 27 degrees.
 29. The assembly of claim 28, wherein the stepratio is 4.122 when the slant angle is in the range of 14 to 15 degrees,is 3.16260 when the slant angle is in the range of 18 to 19 degrees, andis 2.23560 when the slant angle is in the range of 26 to 27 degrees. 30.The assembly of claim 22, wherein the pixels in the ink layer are fixedin the pixel locations and wherein the pixels are configured toconcurrently display the plurality of images.
 31. An optical product,comprising: a transparent lens sheet comprising a first side having aplurality of side-by-side sets of linearly arranged lenses with each setat a slant angle in a range of 10 to 46 degrees from a vertical or ahorizontal axis of the lens sheet; and an image layer, proximate to asecond side of the lens sheet opposite the first side, comprising pixelsfrom a plurality of frames, wherein the pixels are arranged in aplurality of linear patterns of pixel locations with each of thepatterns containing pixels from each of the frames and extendingtransverse and nonorthogonally relative to a line extending through acenter point of one of the sets of the linearly arranged lenses andwherein each of the linearly arranged lenses has one of a round base, asquare base, and a hexagonal base.
 32. The optical product of claim 31,wherein the pattern of pixel locations is adapted to align a number ofthe pixels from each of the digital images to be parallel to a lineextending through a center of the linearly arranged lenses in anadjacent one of the sets of the linearly arranged lenses.
 33. Theoptical product of claim 31, wherein the lens sheet has a thickness inthe range of 10 to 2500 microns and the sets of the linearly arrangedlenses are provided on the first side at 75 to 1500 LPI and wherein eachof the linearly arranged lenses has a size parameter matching an LPI ofthe sets of the linearly arranged lenses.
 34. The optical product ofclaim 31, wherein the plurality of digital images comprises a number ofimages selected from the range of 5 images to 16 images and wherein thelinear patterns include sets of the pixels equal in number to the numberof images including at least one pixel from each of the plurality ofdigital images.
 35. The optical product of claim 34, wherein the sets ofthe pixels providing each of the linear patterns are aligned in a row orin a column of the pattern of pixel locations.
 36. The optical productof claim 34, wherein the slant angle falls within a range of 14 to 15degrees, a range of 18 to 19 degrees, a range of 26 to 27 degrees, or arange of 44 to 46 degrees.
 37. The optical product of claim 36, whereinthe slant angle is 14.04 degrees, 18.435 degrees, 26.57 degrees, or 45degrees.
 38. The optical product of claim 31, wherein a step ratio,defining an increase in an amount of interlaced data when compared withtraditional interlacing, is greater than 4 when the slant angle is inthe range of 14 to 15 degrees, is greater than 3 when the slant angle isin the range of 18 to 19 degrees, and is greater than 2 when the slantangle is in the range of 26 to 27 degrees.
 39. The optical product ofclaim 38, wherein the step ratio is 4.122 when the slant angle is in therange of 14 to 15 degrees, is 3.16260 when the slant angle is in therange of 18 to 19 degrees, and is 2.23560 when the slant angle is in therange of 26 to 27 degrees.
 40. The optical product of claim 31, whereinthe plurality of frames displayed by the pixels in the image layerremain unchanged over time and wherein the pixels concurrently displaythe plurality of frames.